Convergence of the self-avoiding walk on random quadrangulations to SLE$_{8/3}$ on $\sqrt{8/3}$-Liouville quantum gravity

TL;DR

This paper proves that the self-avoiding walk on a random quadrangulation converges to SLE$_{8/3}$ on a Liouville quantum gravity surface, using peeling procedures and Brownian half-plane properties, without requiring prior knowledge of SLE or LQG.

Contribution

It establishes the convergence of the self-avoiding walk on random quadrangulations to SLE$_{8/3}$ on Liouville quantum gravity surfaces, expanding understanding of scaling limits in random geometry.

Findings

Convergence of the SAW on quadrangulations to SLE$_{8/3}$ on LQG surfaces.

Use of peeling procedures for proofs.

Results for both half-plane and whole-plane quadrangulations.

Abstract

We prove that a uniform infinite quadrangulation of the half-plane decorated by a self-avoiding walk (SAW) converges in the scaling limit to the metric gluing of two independent Brownian half-planes identified along their positive boundary rays. Combined with other work of the authors, this implies the convergence of the SAW on a random quadrangulation to SLE$_{8/3}$ on a certain $\sqrt{8/3}$-Liouville quantum gravity surface. The topology of convergence is the local Gromov-Hausdorff-Prokhorov-uniform topology, the natural generalization of the local Gromov-Hausdorff topology to curve-decorated metric measure spaces. We also prove analogous scaling limit results for uniform infinite quadrangulations of the whole plane decorated by either a one-sided or two-sided SAW. Our proof uses only the peeling procedure for random quadrangulations and some basic properties of the Brownian half-plane, so can be read without any knowledge of SLE or LQG.